2.1.

Chemicals

All chemicals were purchased from Sigma Aldrich, Switzerland.

2.2.

Yeast Growing Conditions

Two different growth conditions for yeast cultivation were used in this study. About $2.5\mathrm{mg}/\mathrm{mL}$ Saccharomyces cerevisiae YSC2 (type II) was dissolved either in a standardly used YPD solution ($50\mathrm{mg}/\mathrm{mL}$) or in an aqueous glucose ($150\mathrm{mg}/\mathrm{mL}$) solution.^6,7 Yeast samples were incubated in a thermostabilized (32°C) rotary shaker (210 rpm) for 3 h until further use.

2.3.

Experimental Measurements of the Optical Properties ($R_d$, LPD, and $n$)

The setup shown in Fig. 1 was used to measure the fluence rate and $R_d$ in yeast solutions. An argon ion (457, 488, and 514 nm) and a frequency-doubled YAG (532 nm) lasers (Spectra-Physics, Mountain View, California; Model 2020 and Millennia, respectively) were used as light sources for the measurements. Diode lasers were used for all other wavelengths (typical bandwidth $<5\mathrm{nm}$): 405 nm was generated by a source from Oxxius SA, France, 635 nm was produced by a diode laser from CeramOptec, Germany, whereas 730 and 808 nm were produced by diode lasers provided by Roithner LaserTechnik GmbH, Austria. The light delivered by these light sources was coupled in the SMA connector of a frontal light distributor (Model FD 1, Medlight SA, Switzerland; 2-mm outer diameter) producing a circular (diameter: 15 cm) homogenous spot with sharp edges. The powers delivered by these light distributors ranged between 50 and 500 mW, depending on the detector sensitivity and dynamic ranges.

Fig. 1

Sketch of the experimental setup used to measure the optical properties of the yeast solution models.

Yeast solutions (in glucose or YPD) were prepared according to the protocol described above and placed in a 3-L cylindrical borosilicate vessel with the following geometry: (height: 200 mm; diameter: 150 mm; wall thickness: 3.5 mm). A black velvet mat was put underneath the vessel to absorb the photons reaching the bottom of the vessel. The fluence rate was measured using a scattering spherical isotropic probe (Model IP, Medlight SA, Switzerland; light detection sphere diameter: $850\mu \mathrm{m}$) attached to a translation stage enabling a vertical displacement. This isotropic probe was inserted into a stainless steel tube (outer diameter: 1 mm) to prevent a direct coupling of the light in the optical fiber collecting the light from the scattering sphere. The lower part of this tube was bended at 90 deg to orient optimally the scattering sphere and to prevent possible artifacts induced by the shadow produced by this fiber holder. This tube was covered with a Kodak white reflectance coating (Eastman Kodak Company), a nonfluorescent coating backscattering $>98\%$ of the light between 300 and 1200 nm, to prevent changes of the photon distribution in the solution due to the limited reflectivity of stainless steel. For the measurements at 405 nm, a MV405/20—Narrow Violet Vision Filter (Chroma Technology Corp.) was positioned between the SMA connector of the fiber guiding the light from the isotropic probe and a photosensitive diode (DET 210, ThorLabs) to reject the fluorescence induced by violet light in the yeast solutions.

The fluence rate was measured six times every 5 mm from the top to the bottom of the vessel, and vice versa. The effective attenuation coefficient (${\mu }_{\mathrm{eff}}$) can, in principle, be derived from such measurements by a mono-exponential fit corresponding to the decrease in the fluence rate with depth,^5,16–18 if the illumination is broad enough and if the hypothesis that is at the basis of the diffusion approximation is satisfied (diffuse light flux; ${\mu }_a\ll {\mu }_s^{\prime }$). Although most of these conditions were satisfied in our measurements, edge effects were present in most situations as, in the worst situation, the illumination spot was only about three times larger than ${\mu }_{\mathrm{eff}}^{-1}$. In the other words, the illumination spot could not be considered as semi-infinite. This is the reason why the parameters extracted from our depth measurements of the fluence rate is called “light penetration depth” (LPD) in the following text. The LPD was defined as the inverse of the slope of the linear fit representing the logarithmic decay of the fluence rate measured as a function of depth (Fig. 2). Only the points at which the regression line was within the error bars were considered to determine the slope. This approach was used to reject the points measured close to the surface, where the diffusion approximation does not apply.

Fig. 2

Illustrative measurement of the fluence rate at different depths. Points (in gray) close to the surface were not considered to determine the slope, i.e., only points (in black) at which regression line was within the error bars are considered to determine the slope. Error bars represent the standard deviation of six individual measurements.

The same experimental setup (Fig. 1) was used to assess the total reflectance $R_d$. A scientific CCD camera (Hamamatsu, EM-CCD C9100 equipped with a Fujinon TV ZOOM LENS 1:1.2/12.5-75 mm H6x12.5R - MD 3) was positioned 70 cm above the surface of the yeast solution with an angle of 8 deg relative to the perpendicular to the air–solution surface to minimize the specular reflections. For each picture and wavelength, a region of interest (ROI) was selected while avoiding specular reflections, and the brightness was determined with the software ImageJ (National Institutes of Health, Bethesda, Maryland). The signal acquired in the dark (background) was measured using the same ROI and subtracted from all measurements. At each wavelength, the absolute value of $R_d$ was determined by comparing the brightness obtained with the yeast solutions and when a diffuse reflectance standard (SphereOptics GmbH, Germany), reflecting $>99\%$, was placed at the position corresponding to the yeast solution surface. The refractive index of yeast in glucose or YPD solutions was determined at different wavelengths ranging between 405 and 808 nm at room temperature according to the following procedure. An Abbe’s refractometer (OPL, France) was used to first determine the refractive index of distilled water at 589 nm. Our obtained results were in agreement with those reported in the literature.¹⁹ We then measured the refractive index of the yeast solutions that differed from that of distilled water at this wavelength. Because our refractometer is designed to measure the refractive index at 589 nm only, we estimated the values of this parameter at other wavelengths by adding this difference to the values reported in the literature for water.

2.4.

Monte Carlo Simulation to Determine ${\mu }_a$ and ${\mu }_s^{\prime }$

As mentioned above, if ${\mu }_a$ is much smaller than ${\mu }_s^{\prime }$, these microscopic optical parameters can be calculated from $R_d$ and the LPD using the analytical equation derived in the framework of light diffusion approximation.^16,20 However, as shown below in the results and discussion section, the assumption ${\mu }_a\ll {\mu }_s^{\prime }$ required for the diffusion theory to be valid is not satisfied for the present yeast solutions at all wavelengths. In addition, using the experimental arrangement described above, the effect of the container walls on the light penetration cannot be neglected. Therefore, the optical parameters of the yeast solutions were determined using a Monte Carlo-based ray tracing software (TracePro; Lambda Research Corporation).

The geometry of the modeled system, including the container and the illumination parts, was matched with the experimental setup. On the basis of preliminary studies, the number of photons followed in each simulation was set to $1.2\times {10}^7$. In addition, the Henyey–Greenstein phase function was used to approximate the scattering processes, the corresponding average cosine being set to $g=0.95$. The experimental refractive index values were implemented in the model.

Fifteen virtual spherical probes (diameter 2 mm) separated by 10 mm were positioned along the optical axis inside the yeast solution to monitor the light fluence-rate at different depths. The LPD was calculated from the exponentially decaying part of the fluence-rate spatial distribution, typically in the depth range of 90 to 150 mm (Fig. 2). The reflectance $R_d$ was obtained by counting the photons passing through a circular probe surface (diameter 20 mm) placed 0.1 mm above the solution level at the container center (ROI). To mimic the light collected by the CCD camera in the experimental setup, only the photons propagating in the upward direction at $8\mathrm{deg}\pm 0.5\mathrm{deg}$, with respect to the optical axis, were taken into account. The reference number of reflected photons was obtained when the solutions were replaced by a totally reflecting Lambertian surface predefined in the TracePro software.

For each experimental condition, the optical parameters of the yeast model were determined by an iterative process. After choosing an initial guess for ${\mu }_a$ and ${\mu }_s^{\prime }$, the Gauss–Newton method²¹ was applied consecutively to minimize the sum of squared relative differences (RD)

The use of relative differences compensates for the disproportion of the reflectance and LPD numerical values and ensures the same upper limit for their relative errors. In each iteration step of the Gauss–Newton method, the Jacobian matrix of the RD vector [Eq. (1)] with respect to ${\mu }_a$ and ${\mu }_s^{\prime }$ was determined. This matrix required Monte Carlo simulations at three different (${\mu }_a,{\mu }_s^{\prime }$) pairs. The time needed for a single Monte Carlo simulation was $\sim 45\min$. The iterations were stopped when $S<0.0002$ was reached. It follows that the highest possible difference between the experimental and the converged simulated values of $R_d$ or LPD was 1.4% (the square root of 0.0002). Typically, 3 to 6 iteration steps (each consisting of three simulations) were needed to reproduce one experimental condition (wavelength).

3.1.

Experimental Estimation of the LPD, R_d, and the Refractive Index in Yeast Solutions

The values of ${\mathrm{LPD}}^{-1}$ and $R_d$ assessed experimentally using the setup shown in Fig. 1, which are shown in Fig. 3 for both (glucose and YPD) solutions. The wavelength dependence of these parameters varies significantly between the two solutions, yeast in YPD presenting the shorter penetration depth at short wavelengths. This spectral evolution is probably due to the Soret band of heme derivatives that are more abundant in YPD. The large LPDs (up to 50 mm) measured in the red and near infrared illustrate that our vessel, although relatively large (diameter: 150 mm; depth: 150 mm), cannot be considered as “semi-infinite,” a situation that explains why the optical coefficients have been derived with a Monte Carlo-based simulation. The values of ${\mathrm{LPD}}^{-1}$ and $R_d$ resulting from the optical coefficients derived by the simulation are also shown in Fig. 3.

Fig. 3

Experimental (full signs) and simulated values (open signs) of ${\mathrm{LPD}}^{-1}$ and $R_d$ for yeast in (a) glucose and (b) YPD.

The excellent match between the measured and calculated values of ${\mathrm{LPD}}^{-1}$ and $R_d$ shows the accuracy of the computation method mentioned above.

The refractive indices we have measured at 589 nm and at room temperature were 1.337 and 1.352 for the YPD and glucose solutions, respectively. These values are close to that of pure water and of a $150\mathrm{mg}/\mathrm{mL}$ glucose solution, respectively. This is not surprising as yeast cells are quite diluted ($2.5\mathrm{mg}/\mathrm{mL}$) and consist mostly of water (70%).²²

3.2.

Computational Estimation of ${\mu }_a$ and ${\mu }_s^{\prime }$

The refractive index of both yeast solutions was assessed for other wavelengths, but 589 nm was used as mentioned above to perform the simulations. Our estimated refractive index at other wavelengths in glucose solution led to results that are consistent with the literature.²³

The values we obtained for ${\mu }_a$ and ${\mu }_s^{\prime }$ between 400 and 850 nm are shown in Figs. 4(a) and 4(b) for yeast in glucose and YPD solutions, respectively.

Fig. 4

Estimated optical parameters ${\mu }_a$ and ${\mu }_s^{\prime }$ yeast in (a) glucose and (b) YPD (both black squares); ${\mu }_a$ of distilled water is indicated by the dashed line, whereas the lines presented for ${\mu }_s^{\prime }$ is for visual support only.

It is interesting to note that the absorption at wavelengths $>600\mathrm{nm}$ is mostly due to water. The excellent match between our measurements and the absorption of water in this spectral domain, which contains an overtone of a vibrational band around 750 nm, validates our experimental and data analysis procedure.^24,25 Thus, it can be concluded that in this spectral region, the yeast absorption is negligible. Finally, the wavelength dependence of ${\mu }_s^{\prime }$ is relatively weak, a result that can be expected considering that yeast cells are diluted and relatively large as compared with the wavelength.²⁶